Calculus Examples

$x + \frac{1}{x} + \frac{1}{x + \frac{1}{x}}$

Step 1

Set the denominator in $\frac{1}{x}$ equal to $0$ to find where the expression is undefined.

$x = 0$

Step 2

Set the denominator in $\frac{1}{x + \frac{1}{x}}$ equal to $0$ to find where the expression is undefined.

$x + \frac{1}{x} = 0$

Step 3

Solve for $x$ .

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Step 3.1

Find the LCD of the terms in the equation.

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Step 3.1.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$1, x, 1$

Step 3.1.2

The LCM of one and any expression is the expression.

$x$

Step 3.2

Multiply each term in $x + \frac{1}{x} = 0$ by $x$ to eliminate the fractions.

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Step 3.2.1

Multiply each term in $x + \frac{1}{x} = 0$ by $x$ .

$x \cdot x + \frac{1}{x} x = 0 x$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Simplify each term.

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Step 3.2.2.1.1

Multiply $x$ by $x$ .

$x^{2} + \frac{1}{x} x = 0 x$

Step 3.2.2.1.2

Cancel the common factor of $x$ .

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Step 3.2.2.1.2.1

Cancel the common factor.

$x^{2} + \frac{1}{x} x = 0 x$

Step 3.2.2.1.2.2

Rewrite the expression.

$x^{2} + 1 = 0 x$

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

Multiply $0$ by $x$ .

$x^{2} + 1 = 0$

Step 3.3

Solve the equation.

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Step 3.3.1

Subtract $1$ from both sides of the equation.

$x^{2} = - 1$

Step 3.3.2

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{- 1}$

Step 3.3.3

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \pm i$

Step 3.3.4

The complete solution is the result of both the positive and negative portions of the solution.

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Step 3.3.4.1

First, use the positive value of the $\pm$ to find the first solution.

$x = i$

Step 3.3.4.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - i$

Step 3.3.4.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = i, - i$

Step 4

The domain is all values of $x$ that make the expression defined.

Interval Notation:

$(- \infty, 0) \cup (0, \infty)$

Set-Builder Notation:

${x | x \neq 0}$

Step 5

The range is the set of all valid $y$ values. Use the graph to find the range.

Interval Notation:

$(- \infty, - \frac{5}{2}] \cup [\frac{5}{2}, \infty)$

Set-Builder Notation:

${y | y \leq - \frac{5}{2}, y \geq \frac{5}{2}}$

Step 6

Determine the domain and range.

Domain: $(- \infty, 0) \cup (0, \infty), {x | x \neq 0}$

Range: $(- \infty, - \frac{5}{2}] \cup [\frac{5}{2}, \infty), {y | y \leq - \frac{5}{2}, y \geq \frac{5}{2}}$

Step 7